Thesis title: Formal methods for the design of imitative polyphonic structures Keywords: stacked canon, stretto, fugue, algorithmic composition, counterpoint, counterpointing approach, intervallic approach, harmoniola, rotational array, relative chord tone, dux-graph, graph theory, Eulerian cycle, Hamiltonian cycle, constraint logic programming, chord sequence, chord sequence modulation Abstract This thesis defines novel and efficient methods for the design of stacked canons and their use in imitative polyphonic structures. Chapter 1 discusses the development of canon- and fugue-techniques and their connection in stretto-fugues such as found in Bach’s ‘Kunst der Fugue’. Several examples show that larger polyphonic structures are sustainable by a main theme which can appear in many different canons, called stretti. Hence, techniques to effectively design such themes require the availability of efficient techniques for the creation of several types of canons. In search of such techniques, chapter 2 provides a theoretical basis for the remainder of the thesis. An analysis of the established counterpointing and intervallic approaches to the construction of stacked canons shows that these provide limited harmonic control and are computationally complex. While efficient and in complete control of harmony, Morris’ Tonnetz approach targets serial stacked canons ad minimum and does not encompass voice-leading constraints. A style-independent, constructive approach using relative chord tones, chord sequences and chord sequence modulations is presented along with its connections to graph-theory in address of these issues. An analysis of Rameau’s Canon at the Fifth from his Traité de l’harmonie introduces the concept of relative chord tones and incorporates two constraints discussed in the Traité in the newly proposed relative chord tone model, namely, obtaining complete chords, and, preparation and resolution of sevenths. The analysis explains Rameau’s choice of dux and chord sequence in terms of the conjunction of these constraints. Using my definition of restless dux graphs, the problem of obtaining complete chords is reduced to the Hamiltonian cycle problem in a dux graph. The problem of finding a dux according to the conjunction of the aforementioned constraints is reduced to the definition of a generating function. An approach to the incorporation of voice leading constraints is sketched, by the detailed discussion of prohibitions of firstly parallel octaves, and secondly, parallel fifths. After deriving a least upper bound on the maximum number of voices in a stacked canon without parallel octaves in terms of the canon’s chord sizes, the problem of finding a dux for such a stacked canon with a maximum number of voices is reduced to the Eulerian cycle problem in a restless dux graph. It is also proved that the conjunction of either constraint reduces the aforementioned least upper bound by a single voice. My definition of dux graphs with rests allows the definition of two near linear dux generation algorithms which respectively satisfy the first and either constraint. The problem of constraining the inversion of chords is discussed in terms of the incorporation of a constraint which prohibits 6/4 chords. The remainder of the chapter discusses two methods which allow variations in a stacked canon’s melody based on a predetermined chord sequence, followed by a discussion which counts the number of distinct sub-canons of a stacked canon. It is established that the latter number shows double exponential growth in the chord sizes of a chosen chord sequence, in demonstration of the applicability of my methods to the design of thematic material for use in larger scale imitative polyphonic structures. Chapter 3 discusses my compositions Spiral which is based on several themes, the main two of which were derived as relative chord tone sequences using results from chapter 2. The chapter also discusses several techniques related to the organization of polythematic stacked canonic structures in larger scale composition. The first movement was designed in terms of several harmoniolas, in which neither of the aforementioned themes appears in a stretto. The second movement is designed as a passacaglia structure based on the aforementioned second theme, onto which many distinct combinations of either theme are superimposed. The final movement mirrors the structure of the first movement, testing my conjecture that the use of a chromatic mirror axis can mirror the affects experienced by an audience. Chapter 4 discusses my composition Fugue in G, the main theme of which was derived as a Hamiltonian cycle in the underlying dux graph. The fugue contains many stretti whence the chapter provides further a discussion on the organization of polythematic canonic structures in larger scale composition. While the theory presented in chapter 2 enables one to derive efficient algorithms for the generation of stacked canons adhering to personal stylistic preferences, their derivation may be tedious. This is especially so when experimenting with different rules. Chapter 5 discusses a method which uses constraint logic programs with relative chord tone domains to efficiently search for the thematic material of polythematic stacked canonic structures. The composition process for my ‘Missa ad Fugam’ is discussed in demonstration of this technique. The Mass uses three main themes, each of which allows a four-voice stacked canon, and each pair of which allows a four-voice stacked double canon. Finally, chapter 6 summarizes the main results of this thesis and provides an outlook on future research
